SYSTEM DYNAMICS & CONTROL
CHAPTER 5
SYSTEM STABILITY
EXPLORATION METHODS
Dr. Vo Tuong Quan
HCMUT - 2011
Stability Exploration Methods
Definitions
A stable system is a system with a bounded response to a
bounded input
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Stability Exploration Methods
Examples
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Stability Exploration Methods
Poles and Zeros Method
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Stability Exploration Methods
Poles and Zeros Method
Zeros and Poles plot: a graph which represents the position
of poles and zeros in the complex s-plane
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Stability Exploration Methods
Poles and Zeros Method
The stability of a system depends on the locations of Its poles:
- If all the poles of the system lie in the left-half s-plan then the
system is stable
- If any of the poles of the systems lie in the right-half s-plane
then the system is unstable
- If some of the poles of the system lie in the imaginary axis
and the others lie in the left-half s-plane then the system is at
the stability boundary
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Stability Exploration Methods
Routh Method
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Stability Exploration Methods
Routh Method
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Stability Exploration Methods
Routh Method
The necessary and sufficient condition for a system to be
stable is that all the coefficients of the characteristic equation
are positive and all parameters in the first column of the Routh
table have positive signs.
The number of sign changes in the first column of the Routh
table is equal the number of roots lying in the right-half s-plane.
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Stability Exploration Methods
Routh Method
The system
is stable
because all
the
parameters
in the first
column are
positive
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Stability Exploration Methods
Routh Method
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Stability Exploration Methods
Routh Method
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Stability Exploration Methods
Routh Method
Find the codition of K for the following system to be stable
The characteristic equation of the system
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Stability Exploration Methods
Routh Method
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Stability Exploration Methods
Routh Method
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Stability Exploration Methods
Routh Method
The
system is
Unstable –
Why ??
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Stability Exploration Methods
Roots Locus Method
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Stability Exploration Methods
Roots Locus Method
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Stability Exploration Methods
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Stability Exploration Methods
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